Convergence and comparison theorems for multisplittings
Joan‐Josep Climent, Carmen Perea · Numerical Linear Algebra with Applications · 1999
To solve a linear system Ax = b by an interative method, it is customary to use a splitting of A in the sequential case and a multisplitting of A in the parallel case. In both cases, the convergence of the method is given by the spectral radius of the correspondent iteration matrix. Using the splittings of the second type and establishing an alternative convergence result for weak splittings, we extend the convergence result of O’Leary and White (1985) and the comparison result of Elsner (1989). Also, we introduce new convergence and comparison results for weak multisplittings. On the other hand, introducing the concept of weak nonnegative definite splitting we present new convergence and comparison results for positive definite matrices which extend the results of O’Leary and White (1985) and Nabben (1996). Copyright © 1999 John Wiley & Sons, Ltd.